Asymptotics for orthogonal polynomials and Christoffel functions on a ball
The author investigates orthogonal polynomials in \(d\) real variables on the unit ball of \({\mathbf R}^d\) with respect to the weight function \(W_\mu({\mathbf x})=(1-|{\mathbf x}|)^{\mu-1/2}\). Let \({\mathbf P}_n({\mathbf x})\) be the vector of the orthonormal polynomials of degree \(n=n_1+n_2+\cdots+n_d\), then \({\mathbf P}_n({\mathbf x})\) has \({n+d-1 \choose n}\) components and the author has shown earlier that working with orthogonal polynomials in \(d\) variables is most convenient using this vector notation. This is particularly so in the present paper, where it is shown that for \(d \geq 2\) one can obtain the asymptotic behaviour of \({n+d-1 \choose n}^{-1} {\mathbf P}_n^T({\mathbf x}){\mathbf P}_n({\mathbf x})\) inside the open unit ball, and the limit is the ratio \(W_0({\mathbf x})/W_\mu({\mathbf x})\). Such a limit does not exist for \(d=1\), and in that case one usually investigates the asymptotic behaviour of the Christoffel function, which is \((\sum_{k=0}^n p_n^2(x))^{-1}\). For \(d \geq 2\) one can analyse each term in the Christoffel function separately, and hence, as a consequence the author also obtains the asymptotic behaviour of the Christoffel function, which was obtained earlier by \textit{L. Bos} [New Zealand J. Math. 23, 99-109 (1994; Zbl 0828.42014)]. The proof is based on an explicit formula for \({\mathbf P}_n^T({\mathbf x}){\mathbf P}_n({\mathbf x})\) in terms of an integral involving the Gegenbauer polynomial \(C_n^{\mu+(d-1)/2}\).
orthogonal polynomials in several variables
scientific article; zbMATH DE number 987700